Tri-level cube corner ruling

ABSTRACT

A rulable array of cube corners defined by three sets of parallel, equidistant, symmetrical vee-grooves is provided. The directions of the three vee-groove sets make three angles, no two of which are equal. When the array in viewed in plan, lines along the roots of the grooves determine a pattern of triangles in which the apices of the cube corners lie at distances from their respective triangle&#39;s centroid that are substantially less than the distance between the triangle&#39;s orthocenter and its centroid. In an unmetallized prismatic retroreflective sheeting, the array of quasi-triangular cube corners has the entrance angularity advantages of compound cant with nearly the efficiency of uncanted triangular cube corners at small entrance angles.

This invention relates generally to retroreflective cube corners, andspecifically to an array of quasi-triangular cube corners, ruled bythree sets of equidistant, parallel, symmetrical vee-grooves. The arrayof cube corners achieves improved useful retroreflectance asunmetallized prisms by having the directions of ruling form threeunequal angles, and having the three depths of ruling unequal for eachcube corner.

BACKGROUND OF THE INVENTION

The applicant has observed the following properties of cube corner prismelements that are pertinent to the invention. These observations arediscussed with respect to FIGS. 1-22, all of which areapplicant-generated with the exception of FIG. 21. While the prior artis frequently referred to in this section, the interpretations andobservations of the art are believed to be unique to the applicant.

Retroreflectors return light from a source to the source and its nearvicinity. A retroreflective road sign appears hundreds of times brighterto the driver of a vehicle at night than a plain painted sign. By day,the sign is expected to be about as bright as a plain painted sign. Ifthe sign returned the daytime illumination to its sources, sun and sky,it would be quite dark to the vehicle driver by day. The resolution ofthis paradox is that retroreflective road signs can be effective bothnight and day by failing to efficiently retroreflect light that arrivesfrom some sources while efficiently retroreflecting light from othersources. Retroreflective road sign sheetings are the better for theirability to retroreflect vehicle lights at all their realistic positions,but to not retroreflect vehicle lights at nearly impossible positions.

The position of the illuminating source with respect to the road signsheeting is generally described by two angles: entrance angle β andorientation angle ω. FIG. 1 shows a small rod r perpendicular to a roadsign. Light beam e illuminates the sign. Entrance angle β is the anglebetween e and r. Light beam e casts a shadow s of rod r onto the sign.Entrance angle β could be determined from the length of shadow s.Orientation angle ω is determined by the direction of shadow s. ω is theangle from the nominal “up” direction of the sheeting on the sign toshadow s. The angle is measured clockwise, so in FIG. 1, ω equalsapproximately +65 degrees. If the beam of illuminating light is itselfperpendicular to the sign, then entrance angle β=0 and there is noshadow, so ω has no meaning.

For road sign applications, the cases of large β are almost always caseswhere the sign is approximately vertical but is swiveled to face not inthe direction of the source of illumination. The β values for thesesigns may be greater than 40°. The ω values in these cases are generallyin the range 75° to 95° for signs on the right side of the road or inthe range −75° to −95° for signs on the left side of the road. This isthe basis for the importance of the plus and minus 90° values for ω.

Sheeting having good retroreflectance at large β also at 0° and 180°values of ω has a practical advantage. Sheeting is a roll good and thereare economies to being able to use it either lengthwise or widthwise insign fabrication. Thus the importance of the ω values −90°, 0°, 90° and180°.

Retroreflectors are of two optical kinds. The first kind functions<refract, reflect, refract>. A first, curved refracting surface producesan image of the source on a second surface. The image surface isreflective, either specular or diffuse, so image light returns to thefirst refracting surface, and thence back toward the source. This typeof retroreflector is exemplified by the half-metallized glass spherescomprising many road sign sheetings. The second kind of retroreflectorfunctions <reflect, reflect, reflect>. When light enters a corner wherethree mirrors meet at right angles, it reflects off one, then another,and then the third mirror back towards the source. Likewise when lightenters a prism cut as the corner of a cube it can reflect internally offthe three cube faces and return in its original direction. Light entersand exits through a fourth prism face. The full sequence is <refract,reflect, reflect, reflect, refract>. Since the two refractions are atthe same flat surface, the three reflections are what provide theretroreflection. This type of retroreflector is exemplified by thesub-millimeter prisms ruled in arrays in many road sign sheetings.

The prismatic retroreflector has advantages over the sphericalretroreflector for road signs. Prisms can be packed more efficiently.Prisms can be less aberrant. Prisms can also be more selective aboutwhich source directions to retroreflect and which to not.

“Cube corner element” is defined as a region of space bounded in part bythree planar faces, which faces are portions of three faces of a cubethat meet at a single corner of the cube. The geometric efficiency of aretroreflecting cube corner prism depends on two main factors: effectiveaperture and combined face reflectance. Effective aperture is determinedon the assumption that the second factor is perfect. The geometricalform of the cube corner prism and the refractive index of its materialdetermine how much of the area occupied by the prism can participate inretroreflection for illumination of a particular β and ω. The refractiveindex figures in the effective aperture of the cube corner wheneverβ≠0°, because of the refraction at the entrance/exit surface.

Effective aperture may be found by ray-tracing. Another well-knownmethod of determining effective aperture is illustrated in FIGS. 2A and2B. FIG. 2A shows a view of a cube corner retroreflector in thedirection of a light ray entering the cube corner. For β=0°illumination, this is simply a view of the cube corner normal to thefront surface of the retroreflective sheeting. For any otherillumination, the refraction of the incoming ray at the front surfacemust be taken into account when applying the method of the FIG. 2A.

FIG. 2A diagrams the kind of ray path that is required for a cube cornerretroreflection. Illumination enters the cube corner on a ray and thecube corner is pictured as if viewed along that ray. The ray appears aspoint A in FIG. 2A. The ray reaches a point on face 1 shown at A. Theray then reflects from face 1 to another cube face. The path of thisreflection, in the view of the Figure, must appear parallel to the cubedihedral edge 4, which is the dihedral edge that is not part of face 1.The reflected ray reaches face 2 at the point shown at B. Point B isconstructed by making dihedral edge 6, the dihedral edge shared by faces1 and 2, bisect line segment AB. From the point in face 2 shown at B theray reflects to cube face 3. The path of this reflection, in the view ofthe Figure, must appear parallel to the cube dihedral edge 6, which isthe dihedral edge that is not part of face 3. The reflected ray reachesface 3 at the point shown at C. Point C is constructed by makingdihedral edge 4, the dihedral edge shared by faces 2 and 3, bisect linesegment BC. From the point in face 3 shown at C, the ray leaves the cubecorner in a direction parallel to its first arrival, accomplishingretroreflection. This ray appears as point C.

The shaded region of FIG. 2B shows what area of the cube corner of FIG.2A is optically effective for retroreflecting illumination that has thedirection of the view. This effective aperture is the collection of allpoints like point A, as described above, for which there is a point B,as described above, on a second face of the cube corner and also a pointC, as described above, on a third face of the cube corner.

The cube corner apex appears at point O of FIG. 2A. By geometry, AOC isstraight and AO=OC. The illumination entry point and the illuminationexit point are symmetrical about the apex, in the diagram. For cubecorners of triangular shape, with the three dihedral edges eachextending to a triangle vertex, it can be proved that whenever suchsymmetrical points A and C lie within the triangle the intermediatepoint B must also lie within the triangle. Thus the diagrammatic methodof FIG. 2A simplifies for triangular cube corners. The effectiveaperture can be found as the intersection of the cube corner triangleand this triangle rotated 180° about point O, as shown in FIG. 2C.

In determining the effective aperture by the diagrammatic method of FIG.2A, faces are assumed to reflect like mirrors. Faces of a cube cornerprism that are metallized do reflect like mirrors, although there issome loss of intensity by absorption at each reflection. Formetallization with vacuum sputtered aluminum the loss is about 14%. Theloss due to three such reflections is about 36%. Faces of a cube cornerthat are unmetallized may also reflect like ideal mirrors. Totalinternal reflection (TIR) involves zero loss in intensity. However,faces of a cube corner that are unmetallized may also reflect feebly.TIR requires that the angle of incidence upon the face exceed a certaincritical angle. The critical angle is equal to the arc sine of thereciprocal of the refractive index of the prism material. For example,for n=1.5 material the critical angle is about 41.81°. Light incident at41.82° is totally internally reflected. Light incident at 41.80° loses11% of its intensity. Light incident at 41° loses 62% of its intensity.A cube corner prism with unmetallized faces may have one or two facesfailing TIR for a particular incoming illumination.

The geometric efficiency of a retroreflecting cube corner prism dependsalso on the specular reflectance of the surface through which the lightenters and exits the prism. This factor depends on the refractive indexof the front surface material according to Fresnel's equations fordielectric reflection. The front surface material is often differentfrom the prism body material. This factor in geometric efficiency willbe ignored here, since it is independent of the prism design.

Geometric efficiency of an array of retroreflecting cube corners is notcompletely determined by the geometric efficiency of the individualprisms. Some light that is not retroreflected by one prism can travel toother prisms, generally via total internal reflection off the frontsurface, and certain routes involving multiple cube corners produceretroreflection. This factor depends on the prism design as well as onthe thickness of material between the prisms and the front surface. Thisfactor is best studied by raytracing. Since the cube corners of thepresent invention do not differ greatly in their inter-cube effects fromprior art cube corners, this factor is ignored in the descriptions.

The first factor in a cube corner prism's geometric efficiency, itseffective aperture for a particular β and ω, is independent of its beingmetallized or not. The second factor in a cube corner prism's geometricefficiency, the product of the reflectances of its three faces for aparticular β and ω, is greatly dependent on metallization or not.Typically, the combined face reflectance of aluminized cube cornerprisms is about 64% with little dependence on β and ω. Typically, thecombined face reflectance of non-metallized cube corner prisms is 100%for many β, ω combinations and less than 10% for many other β, ωcombinations.

Retroreflective sheeting for road signs must not only retroreflectheadlights at night but also have good luminance by day. Non-metallizedcube corners are always preferred over metallized cube corners for roadsign applications, because metallized cube corner sheeting appearsrather dark by day. This darkness to the vehicle driver is in large partdue to the combined face reflectance of the metallized prisms neverbeing low, so they can better retroreflect sunlight and skylight back tosun and sky. By comparison, TIR frequently fails in a non-metallizedprismatic sheeting, and then light leaks out of the prism. A whitebacking film behind the prisms diffusely reflects such light toeventually emerge nearly diffusely from the sheet.

K. N. Chandler, in an unpublished research paper for the British RoadResearch Laboratory entitled “The Theory of Corner-Cube Reflectors”,dated October, 1956, charted the TIR limits for some non-metallized cubecorner retroreflectors. FIG. 3 shows a square cube corner with fourillumination directions labeled −90, 0, 90, 180 according to angle ω.Shape is irrelevant to TIR limits, which depend solely on the tilts ofthe cube faces and the refractive index of the material.

FIG. 4 shows a diagram corresponding to FIG. 3, in the manner ofChandler's diagram. In FIG. 4, orientation angle ω (from −180° to 180°)is represented circumferentially and entrance angle β (from 0° to 90°)is represented radially, and the three-branched curve shows the maximumβ at each ω for which TIR is maintained in the cube corner. A refractiveindex, n=1.586, was chosen for the example. FIG. 4 shows how theilluminating rays indicated +90 in FIG. 3 can be at any obliquitywithout there being TIR failure, while the light indicated −90 in FIG. 3cannot exceed approximately 25° obliquity without TIR failure, and thelight indicated 0 and 180 in FIG. 3 cannot exceed approximately 31°without TIR failure.

To alleviate the weakness at ω=−90° shown in FIG. 4, cube corners werecommonly paired as shown in FIG. 5. The two Chandler curves in FIG. 6,corresponding to the two cube corners in FIG. 5, show how the left cubein FIG. 5 “covers for” the right cube in FIGS. 3 and 6 at omega=−90°,with the opposite occuring at omega=+90°. However, FIG. 6 shows noimprovement beyond the 31° entrance angle at omega=0°.

The Chandler diagram depends on just two things: the angles at which theinterior rays meet the three cube faces and the critical angle of theprism material. Thus the only non-trivial methods for changing theChandler diagram are change of the refractive index of the prismmaterial and tilt of the cube corner with respect to the article frontsurface.

Increasing the refractive index swells the region of TIR in the Chandlerdiagram as shown in FIG. 7. To shift the directions of the region's armsrequires canting the cube corner.

A cube corner prism element in sheeting is said to be canted when itscube axis is not perpendicular to the sheeting front surface. The cubeaxis is the line from cube apex making equal angles to each of the threecube faces. This line would be a diagonal of the complete cube. RowlandU.S. Pat. No. 3,684,348 discloses “tipping” triangular cube corners inorder to improve their large entrance angle performance at the expenseof their small entrance angle performance. When an array of cube cornersis formed by three sets of parallel symmetrical vee-grooves, and thedirections of grooving are not at 60° to one another, the cube cornersare canted.

Heenan et al. U.S. Pat. No. 3,541,606 discloses non-ruled canted cubecorners with attention to the direction of cant. He found that aretroreflector comprised of unmetallized hexagonal cube corners andtheir 180° rotated pairs could have extended entrance angularity in twoorthogonal planes (i.e., at ω=−90°, 0°, 90°, and 180°) provided the cubecorners were canted in a direction that make a cube face more nearlyparallel to the article front surface. This effect was due to 100%combined face reflectance at large β for these ω values. FIG. 8 shows aplan view of a pair of 10°canted cube corners like those of FIG. 19 ofHeenan et al. U.S. Pat. No. 3,541,606, except for being square ratherthan hexagonal. The cube axes, shown as arrows, show that the cantinghas been symmetrical between the two cube faces that were not made moreparallel to the article front. The dihedral edge between those twofaces, the cube axis, and a normal line from the cube apex perpendicularto the article front surface lie in a single plane, and said normal linelies between said dihedral edge and the cube axis.

Applicant has found it useful to construct diagrams like Chandler's, butfor canted cube corners. In this application all such diagrams arecalled “Chandler diagrams”. FIG. 9 is the Chandler diagram for the FIG.8 pair of “face-more-parallel”, abbreviated “fmp”, cube corners formedin acrylic. FIG. 10 is the Chandler diagram for acrylic cube cornerswith fmp cant greater by 1.3°.

Hoopman U.S. Pat. No. 4,588,258 discloses applying the fmp cant to ruledtriangular cube corners. The Chandler diagram generated for such ruledtriangular cube corners is substantially the same as that obtained forHeenan et al. U.S. Pat. No. 3,541,606. Hoopman's cube corners have evenbetter entrance angularity than Heenan's like canted cube cornersbecause the triangular cube corners have greater effective aperture atlarge entrance angles than the hexagonal or square cube corners.

Heenan et al. U.S. Pat. No. 3,541,606 also discloses the“edge-more-parallel” canted cube corner, abbreviated “emp”. FIG. 11shows a pair of such cubes with 10° emp cant. The cube axes, shown asarrows, show that the canting is symmetrical between two cube faces insuch a way that the dihedral edge between them becomes more parallel tothe article front surface. Said dihedral edge, the cube axis, and anormal line from the cube apex perpendicular to the article frontsurface lie in a single plane, and the cube axis lies between saiddihedral edge and said normal line. FIG. 12 is the Chandler diagramgenerated for acrylic n=1.49 cube corners of FIG. 11. FIGS. 9 and 12show the respective earmarks of face-more-parallel andedge-more-parallel cant. The symmetrical Chandler diagram from FIG. 6 iscompressed in FIG. 9 and stretched in FIG. 12. Comparing FIGS. 6, 9, and12, the diagramatic area of TIR is greatest in FIG. 6 and least in FIG.9. However, the TIR region in FIG. 9 contains the most useful β,ω pairs.

Smith et al. U.S. Pat. Nos. 5,822,121 and 5,926,314 disclosed the rulingof arrays of cube corners by means of three sets of parallel symmetricalvee-grooves to equal depth, the grooves having directions such thatbetween no two are the angles the same. The cube corners have the shapesof scalene triangles. Applicant has observed that the cube axes arenecessarily canted, but the cant is neither flip nor emp. FIG. 14 showsa plan view of a pair of such cube corners with cant 9.74°. For eachcube, the cube axis, shown as an arrow, show that the canting is notsymmetrical between any two cube faces. There is no dihedral edge in aplane together with the cube axis and a normal drawn from the cube apexto the article front surface. In this application, such cant is called“compound cant”.

FIGS. 13A and 13B explain the shape of the Chandler diagram for acompound canted cube corner. FIG. 13A is the same plan view, normal tothe article front surface, of one of the cube corners of FIG. 14, butthe heavy arrows are different from the arrows of FIG. 14. The heavyarrows FIG. 13A follow the altitudes of the triangle and indicate theillumination orientation angles which, for a given entrance angle, makethe smallest incidence angles upon the cube faces. For a given entranceangle, for all orientation angles of illumination, that along the arrowmarked a will reach the face marked a at the smallest incidence anglebecause it alone has just one dimension of obliquity. Thus TIR will failat face a for this orientation angle at a smaller entrance angle than atother orientation angles. FIG. 13B is the Chandler diagram for the cubecorner of FIG. 13A. The arrow marked a in FIG. 13B corresponds to thearrow marked a in FIG. 13A. Arrow a in FIG. 13B points to the minimum βon that arcuate portion of the Chandler diagram that indicates whereface a of FIG. 13A fails TIR. The arcuate portion is symmetrical aboutarrow a.

Applicant has found that if one edge of the triangle cube corner is madeupright as in FIG. 13A, then if the triangle has angles A and B on thatedge as shown in the Figure, elementary geometry determines that theChandler diagram will have its three limbs centered at approximately thethree χ angles:ω₁=90°−A−B;ω₂=90°+A−B;ω₃=90°+A+B;  (1)

For the example of FIG. 13A, angle A=50° and angle B=60°, so the threeChandler limbs are centered on approximately −20°, 80°, and 200°. Ofgreater importance are the three angles separating the three limbdirections. These are approximately:Δω₁=2A;Δω₂=2B;Δω₃=360°−2A−2B,or Δω₃=2C, where C is the third angle of the triangle.  (2)

It is desirable to have two limbs about 90° apart. According to theabove relations, this requires one of the plan view triangle angles toequal 45°. Applicant has observed that this is not possible with aface-more-parallel canted isosceles triangular cube corner, since thetriangle would be 45°-45°-90° implying that the plan view is squarelyupon one face. Limbs 100° apart is good enough. This requires thetriangle to be 50°-50°-80° which implies a cant of about 21.8°. Theconsequence of such large cant is a failure of TIR at β=0°. FIG. 16shows how even a 16° face-more-parallel cant, with even a very highrefractive index n=1.63, nearly fails TIR at β=0°.

Applicant has further observed that Chandler limbs 90° apart is possiblewith a edge-more-parallel canted isosceles triangular cube corner, bymaking the triangle 67.5°-67.5°-45°. This corresponds to approximately10.8°cant. FIG. 12 shows the Chandler diagram for nearly this cubecorner. There are problems involving the effective aperture at large βfor emp designs, as will be discussed later.

Chandler limbs 90° apart is possible with a scalene triangular cubecorner, such as the one with A=45°, B=60°, C=75°. It is more practicalto make limbs 100° apart with A=50° as in FIG. 13A.

FIG. 14 illustrates the cube corner of FIG. 13A with a neighbor cubecorner. Dashed arrows indicate the cube axes in the plan view. FIG. 15illustrates how the Chandler diagrams for the two cube corners cover foreach other. Rotating these cube corners approximately 10°counter-clockwise rotates the Chandler diagram likewise. Then there ispossibility of good entrance angularity for ω=−90°, 0°, 90°, 180°.

FIG. 15 more resembles FIG. 9 than it resembles FIG. 12. In FIGS. 15 and10, the six Chandler limbs are beginning to converge to four limbs.Applicant has observed that it can be shown that in general the sixlimbs are spaced according to 180°−2A, 180°−2B, and 180°−2C. Thus limbconvergence is a result of one of the angles A, B, C being especiallylarge. The isosceles triangle edge-more-parallel cube corner cannot haveany angle especially large since its two largest angles are equal.

FIGS. 17A-17F are plan views of cube corners seen normal to the frontsurface of the sheeting, and the corresponding Chandler diagrams forpaired cube corners. All the cube corners are canted by 11.3°, with theaxis in plan view shown as a short arrow. The figures illustrate thecontinuum of cants from face-more-parallel of 17A to theedge-more-parallel of 17F. The isosceles triangle of FIG. 17A passesthrough scalene triangles to the isosceles triangle of FIG. 17F.Applicant has observed that in FIG. 17A, one face, indicated mp, isespecially vulnerable to TIR failure because it is especially parallelto the sheeting front surface. In FIG. 17F, two faces, each indicatedmp, are vulnerable to TIR failure because they are especially parallelto the sheeting front surface. The two faces flank the edge that iscanted more parallel to the sheeting front surface. It is artificial toclassify all cants as either face-more-parallel or edge-more-parallel aswas attempted in Heenan et al. U.S. Pat. No. 6,015,214, since theoptical characteristics must change continuously betweenface-more-parallel and edge-more-parallel.

Ruled triangular cube corners are useful for illustrating the continuumbetween face-more-parallel and edge-more-parallel cant, but cube cornercant is independent of cube corner shape. Cant is evident from the planview, perpendicular to the sheeting surface, of the three angles formedat the cube apex. If D and E are two of the three angles formed aroundthe apex in this view, and if d=−tan D and e=−tan E, then the cant isgiven by equation (3) which is equivalent to an equation in Heenan etal. U.S. Pat. No. 6,015,214. $\begin{matrix}{{c\quad a\quad n\quad t} = {\arccos\left( {\frac{1}{\sqrt{3d\quad e}}\left\lbrack {1 + {\left( {\sqrt{d} + \sqrt{e}} \right)\sqrt{\frac{{d\quad e} - 1}{d + e}}}} \right\rbrack} \right)}} & (3)\end{matrix}$

For ruled triangular cube corners, the triangle's three angles aresimply the supplements of the angles in the plan view about the cubeapex. For example, in FIG. 17C, the angle marked A plus the angle markedD must equal 180°.

Applicant provides the following five definitions of terms about cubecant:

Cube axis: The diagonal from the corner of a cube, said cube and itscorner underlying the cube corner element.

Canted cube corner: a cube corner having its axis not normal to thesheeting surface. Cant is measured as the angle between the cube axisand the sheeting surface normal. Comment: when there is cant, a planview normal to the sheeting surface shows the face angles at the apexnot all 120°.

Edge-more-parallel cant: cube corner cant such that the cube axis, oneof the dihedral edges, and a normal from the cube corner apex to thesheeting surface lie in one plane and the normal is between the cubeaxis and the dihedral edge. Comment: when cant is emp, a plan viewnormal to the sheeting surface shows two of the face angles at the apexequal, and smaller than the third face angle at the apex.

Face-more-parallel cant: cube corner cant such that the cube axis, oneof the dihedral edges, and a normal from the cube corner apex to thesheeting surface lie in one plane and the dihedral edge is between thecube axis and the normal. Comment: when cant is fmp, a plan view normalto the sheeting surface shows two of the face angles at the apex equal,and larger than the third face angle at the apex.

Compound cant: cube corner cant such that the cube axis, one of thedihedral edges, and a normal from the cube corner apex to the sheetingsurface do not lie in one plane. Comment: when there is compound cant, aplan view normal to the sheeting surface shows no two of the face anglesat the apex equal.

Arrays of cube corners defined by three sets of parallel symmetricalvee-grooves ruled to equal depth are triangular cube corners. For thesecube corners the triangle shape determines the cant and the cantdetermines the triangle shape. Cant is indicated by the angles in theplan view about the cube apex. Applicant has made the followingobservations with respect to cant and effective aperture. Cantdetermines, in conjunction with the index of refraction of the prismmaterial, the effective aperture for each β,ω pair. FIGS. 18A shows anuncanted triangular cube corner and FIGS. 18B-D show three differenttriangular cube corners each having 9.74° of cant. Effective aperturesare indicated for β=0°, at which angle the refractive index has noeffect. The 9.74° canted cube corners have from 50% to 53.6% effectiveaperture at β=0°, compared with 66.7% for the uncanted cube corner. Thetriangles in FIGS. 18A-D are drawn with equal areas. When expressed as afraction or percentage, “effective aperture” means the area of the cubecorner that can participate in retroreflection divided by the area thatthe cube corner occupies in the array.

Through either geometric construction or by ray tracing, the effectiveaperture may be determined for arbitrary beta and omega. FIGS. 19A-Fillustrate applicant's observations as to how the effective aperture ofsome triangular prism cube corners, refractive index 1.586, changes withβ for four different ω's:

−90°; 0°; 90°; 180°; FIG. 19A for cant 0°; FIG. 19B for cant 9.74° fmp;FIG. 19C for the compound 9.74° from a 50°-60°-70° triangle; FIGS. 19Dand 19E for 9.74° emp. FIGS. 19A-F are each for a single cube corner.There is an area of sheeting front surface corresponding to the wholecube corner prism. The calculation of fractional or percent effectiveaperture is on the basis of this area projected in the direction ofillumination, that is, multiplied by the cosine of beta.

FIGS. 20A, 20B, and 20C are Chandler diagrams and plan views for thecanted cube corners from FIGS. 19B, 19C, and 19D&E respectively. Thetriangular cube corner of FIG. 20B has no symmetry plane so it isunobvious how to define 0° or 90° omegas. The cube corner was rotated tomake the thickest limb of its Chandler diagram center on 90° omega. Notethat the entrance angularity is large near 180° omega but not near 0°omega. The pair cube, which takes care of −90° omega also takes care of180°. This exploiting of asymmetry is the trick for improved entranceangularity with such cube corners.

All the curves in FIG. 19A for the uncanted cube corner show effectiveapertures decreasing with increasing beta. Each of FIGS. 19B-F, for thecanted cube corners, have at least one curve showing effective apertureinitially increasing with increasing beta. The 9.74° fmp cube has thisfor omega=90° (FIG. 19B). The 50-60-70 cube has this for omega=0° and90° (FIG. 19C). The 9.74° emp cube has this for −90° (in FIG. 19D).

Chandler diagrams indicate for which β and ω values the combined facereflectance is high. This is a necessary, but not sufficient condition,for high retroreflectance. The other factor is effective aperture.Comparing FIGS. 19A-E and corresponding FIGS. 20A-C allows quickappraisal of designs. In particular, comparing FIGS. 19D and 19E withFIG. 20C reveal problems with the 9.74° emp cube corner. The Chandlerdiagram limits beta to just 42.8° for omega=−90°. Thus the highesteffective apertures shown in FIG. 19D are wasted. The Chandler diagramshows unlimited beta for omega=+90°, where FIG. 19D shows a weakeffective aperture curve. The Chandler diagram shows unlimited beta alsofor omegas −45° and −135° (indicated as 225°). FIG. 19E shows a weakeffective aperture curve for these two omegas. FIG. 19E shows a strongeffective aperture curve for omegas +45° and +135°, but FIG. 20C showsthat TIR is limited to beta=19.7° in those directions. The 9.74° empcube corner is a dunce among canted cube corners for suchdiscoordination of the two factors.

The 9.74° fmp canted triangle cube corner has better luck. Its highcurve of effective aperture in FIG. 19B is for omega=+90°, for whichomega the cube, according to FIG. 20A, sustains TIR throughout theentrance angles. Its middling curves for effective aperture in FIG. 19Bare for omega=0° and 180°, which show middling TIR sustenance in FIG.20A. Its weakest curve for effective aperture in FIG. 19B is foromega=−90°, for which angle TIR is strongly truncated according to FIG.20A. The omega=−90° will be covered by the mate cube. Hoopman U.S. Pat.No. 4,588,258 discloses fmp canted triangular cube corner pairs having abroad range of entrance angularity for all four omegas: −90°; 0°; 90°;180°. Applicant has observed that this is due to the advantageouscoordination of the two geometrical factors.

Applicant has observed that the most harmonious interaction of the twogeometrical factors occurs for the compound canted triangle cube cornerexemplified by the 50°-60°-70° prism of FIGS. 19C and 20B. As with thefmp canted cube corners of FIGS. 19B and and 20A, the highest curve ofFIG. 19C is for omega=+90°, for which omega the cube, according to FIG.20B, sustains TIR throughout the entrance angles. Also corresponding tothe fmp canted cube example, the weakest curve of FIG. 19C is foromega=−90°, for which angle TIR is strongly truncated according to FIG.20B. The compound canted cube corner differs from the fmp cube corner inthat FIG. 19C has separate curves for omega=0° and omega=180°,respectively low and high, while FIG. 19B has a single middling curve.FIG. 20B shows that TIR is truncated at β=34.9° for the omega=0°direction while TIR is sustained to β=72.9° for the omega=180°. There isbeautiful coordination between FIGS. 19C and 20B. The omega=−90° andomega=0° directions will be covered by the mate cube. Smith et al. U.S.Pat. Nos. 5,822,121 and 5,926,314 disclose scalene triangular cubecorner pairs having a broad range of entrance angularity for all fouromegas: −90°; 0°; 90°; 180°.

Retroreflective sheetings must be thin to be flexible, so the cubes mustbe small, on the order of 30 μm to 150 μm deep. Cubes of this sizediffract light within a spread of angles relevant to roadwayperformance. Thus diffraction analysis of sheeting cube optical designsis necessary. Small active areas imply large diffraction patterns. Ingeneral, a design in which one of a pair of cubes sustains large activearea, while the other dies, is preferable to a design in which both of apair of cubes sustain middling active areas, totaling as much as thefirst design. For this and other reasons given, the compound cantedtriangle cube corner prism is advantageous over the fmp and emp types.

FIGS. 18A-D and again FIGS. 19A-E show that the effective aperture forthe canted eamples for β=0° is between three-fourths and four-fifths ofthe effective aperture of the uncanted cube corner for β=0°. Since amajority of road sign uses have β always near 0°, this is a seriousdefect of the canted examples. The defect at β near 0° can be reduced bycanting much less, that is, by compromising with the uncanted cubecorners. Thus Szczech U.S. Pat. No. 5,138,488 discloses the performanceof 4.3° fmp canted cube corner prisms. However 4.3° of cant, withpairing, with moderate refractive index such as 1.586, is too littlecant to provide large entrance angularity in all four omega directions:−90°, 0°; 90°; 180°.

FIG. 21 is identical to FIG. 31 of Heenan et al. U.S. Pat. No.6,015,214. It shows a two part tool comprising a non-rulable array oftriangular cube corners. The tool would be repeated many times,adjoining at faces like that marked 124, for making a full tool. Thefront surface of the sheeting produced will be perpendicular to thelines shown vertical in FIG. 21. Supposing that the triangular bases inFIG. 21 are equilateral, and supposing that angle x equals 9.74°, thenthe cube corners are alternately canted 9.74° fmp and 9.74° emp. Howeverthey do not look like the ruled triangles with corresponding cants inFIGS. 18B and 18D. FIG. 22 shows a plan view, normal to the sheeting, ofthe alternating fmp and emp cube corners, that would result from thetool of FIG. 21. Each cube corner has effective aperture of 62.7% atβ=0°. This compares favorably to 50.0% for the ruled 9.74° fmp cubecorner of FIG. 18B and also to 53.6% for the ruled 9.74° emp cube cornerof FIG. 18D. Heenan et al. U.S. Pat. No. 6,015,214 did not disclose orsuggest these advantages in effective apertures for the triangular cubecorners of FIG. 21. The first advantage should be understood as ageometrical consequence of making the singular edge of the fmp triangleless deep than the rest of the triangle. The second advantage should beunderstood as a geometrical consequence of making the singular edge ofthe emp triangle deeper than the rest of the triangle. Depth is regardedviewing downward upon the tool in FIG. 21.

Mimura et al. U.S. Pat. Nos. 6,083,607 and 6,318,866 B1 disclose that ifthe ruling of emp triangular cube corners is modified to make the sharpgroove, corresponding to the isosceles triangle's short edge, deeperthan the other two grooves, this generally improves the effectiveaperture. Mimura et al. U.S. Pat. No. 6,390,629 B1 discloses that ifruling of fmp triangular cube corners is modified to make the bluntgroove, corresponding to the isosceles triangle's long edge, deeper thanthe other two grooves, this generally improves the effective aperture.

Hexagonal or rectangular cube corners generally have 100% effectiveaperture at β=0°, which falls rapidly with increasing β. Heenan et alU.S. Pat. No. 6,015,214 discloses decentering the apex in a hexagonal orrectangular cube corner in order to improve retroreflectance at large βwhile sacrificing retroreflectance at small β Decentering the apex doesnot affect the Chandler diagram, but strongly affects the effectiveaperture for various β and ω. Triangular cube corners have relativelysmall effective aperture for small β. An uncanted triangle cube cornerhas only 66.7% effective aperture for β=0°. More desirably cantedtriangle cube corners are weaker yet for β=0°.

The purpose of this invention is to improve the effective apertures ofthe most desirably canted triangular cube corners. The above discussionhas identified these as triangular cube corners having compound cants,rather than the face-more-parallel or edge-more-parallel cants. Thetechnique for improving the effective aperture involves ruling thedefining grooves to three different depths so as to displace the apex ofthe cube corner towards the centroid of the triangle describing thegroove paths, all as seen in plan view.

SUMMARY OF THE INVENTION

The invention encompasses an array of cube corner elements defined bythree sets of parallel vee-grooves, the directions of the three groovesets making three angles no two of which are equal. Additionally, whenthe array is viewed in plan in a direction normal to the array, linesalong the roots of the grooves determine a pattern of triangles in whichthe apices of the cube corners lie at distances from their respectivetriangle's centroid that is substantially less than the distance betweenthe triangle's orthocenter and its centroid. The cube corners may bemale, micro-cube corners formed in a transparent material and having asecond substantially flat surface, and so may be retroreflective oflight entering the second surface. The cube corners of the invention areintended to be unmetallized prisms.

In one embodiment of the invention, the roots of the three sets ofvee-grooves lie in three distinct parallel planes. In another embodimentof the invention, the three sets of vee grooves do not lie in threeparallel planes, but the grooves of one set are generally deeper thanthose of a second set which are generally deeper than those of the thirdset, depth being with respect to a reference plane. In one embodiment,the depths in the shallow set are no more than 90% of the depths in theintermediate set, which are no more than 90% of the depths of the deepset.

In each embodiment, the typical cube corner is defined by threevee-grooves having different included angles. The bluntest vee-groove isat the shallowest depth, the intermediate vee-groove is at theintermediate depth, and the sharpest vee-groove is at the greatestdepth.

It is because the groove directions make no two angles equal that thecube corner has a desirable compound cant, and because the groove depthsare made unequal in the manner described that the geometric efficiencyclosely approaches that of uncanted triangular cube corners at thesmaller entrance angles. Varying the three groove depths can displacethe cube apex anywhere within the triangle formed by the lines of thegroove roots, all in plan view. With equal groove depths, the apexappears at the triangle's orthocenter. The invention considers in detailthe efficiencies of displacing the apex along the line from thetriangle's orthocenter to its centroid. Geometric efficiency easilyexceeds 60% for 0° entrance angle over most of the line segment.Excellent efficiencies for a wide range of entrance angles, at theneeded orientation angles, without excessively deep cuts, are achievedwhen displacement is between ¼ and ¾ of this line segment. A preferredembodiment has the apex displaced half way from the orthocenter to thecentroid.

The basic inventive cube corner arrays can produce very high entranceangularity in the four most important directions ω=−90°, 0°, 90°, 180°.For improved symmetry, but still with emphasis on these four ω values,these arrays can be “pinned”, assembling sub-arrays with like sub-arraysdiffering by 90° of rotation, or by mirroring.

Finally, the use of three sets of parallel, equidistant, symmetricalvee-grooves to define the shapes of the inventive cube corners allowsthe master tools to be made easily and accurately, and the simplicity ofthe overall topographic structure allows plastic retroreflectivesheeting to be made efficiently and cost effectively.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a perspective view of a road sign having an imaginary rodprojecting from its center to illustrate the entrance angle β andorientation angle ω;

FIG. 2A is a view of a cube corner in the direction of entering lightillustrating how the exit point C is determined from the arrival pointA;

FIG. 2B illustrates the area of the cube corner of FIG. 2A that isoptically effective for retroreflection;

FIG. 2C illustrates how the effective aperture of a triangular cubecorner can be found as the intersection of the cube corner trianglerotated 180° about the apex point;

FIG. 3 is a plan view of a prior art uncanted cube corner, illustratingthe different omega directions from which an incident beam of light canenter;

FIG. 4 is a Chandler diagram for the prior art cube corner of FIG. 3;

FIG. 5 illustrates how uncanted cube corners have been “paired” in theprior art in order to provide large entrance angle TIR over a range oforientation angles ω that includes −90° and +90°;

FIG. 6 illustrates the superimposed Chandler diagrams for the two cubecorners of FIG. 5, as acrylic prisms;

FIG. 7 illustrates how the Chandler diagram for an uncanted prismaticcube corner changes as a function of the index of refraction of itsmaterial;

FIG. 8 is a plan view of a pair of prior art 10° face-more-parallelcanted cube corners;

FIG. 9 is a Chandler diagram for the cube corners illustrated in FIG. 8,as acrylic prisms;

FIG. 10 is a Chandler diagram for a pair of prior art cube corners suchas those illustrated in FIG. 8 but with 11.3° face-more-parallel cant,as acrylic prisms;

FIG. 11 is a plan view of a pair of prior art 10° edge-more-parallelcube corners;

FIG. 12 is a Chandler diagram for the cube corners illustrated in FIG.11, as acrylic prisms;

FIG. 13A is a plan view of a triangular cube corner of the prior art,wherein the heavy arrows follow the altitudes of the triangle andindicate the illumination orientation angles that, for a given entranceangle, make the smallest incidence angles upon the cube faces;

FIG. 13B is the Chandler diagram for the cube corner of FIG. 13A whereinthe arrows correspond to the arrows in FIG. 13A;

FIG. 14 is a plan view of a pair of FIG. 13A prior art cube cornersillustrating the skew relationship between the cube axes;

FIG. 15 is the Chandler diagram for the pair of cube corners illustratedin FIG. 14;

FIG. 16 is the Chandler diagram for 16° face-more-parallel prism cubecorners of n=1.63 material;

FIGS. 17A-17F illustrate the continuum of 11.3° cants fromface-more-parallel, through compound, to edge-more-parallel, and thecontinuum of Chandler diagrams for the corresponding acrylic prisms;

FIGS. 18A-18D illustrate how the effective aperture of prior art ruledtriangular cube corners at normal incidence vary with the cant;

FIGS. 19A-19E illustrate how effective aperture varies as a function ofentrance angle β at selected ω's for prior art ruled triangularpolycarbonate cube corners without cant and canted 9.74°face-more-parallel, 50-60-70 compound, and 9.74° edge-more-parallel;

FIGS. 20A, 20B, and 20C are the Chandler diagrams for the 9.74° cantedcube corners of FIGS. 19B, 19C, and 19D&E, respectively;

FIG. 21 is a perspective view of a prior art male tool comprisingnonrulable triangular cube corners;

FIG. 22 illustrates the effective apertures of two cube corners fromFIG. 21.

FIG. 23 illustrates theorem 1 in the specification;

FIG. 24 illustrates theorem 2 in the specification;

FIG. 25 illustrates how for a non-equilateral triangle, the orthocenteris distinct from the centroid;

FIGS. 26A and 26B illustrate how the apex of a cube corner defined by ascalene triangle might be favorably displaced by tri-level ruling;

FIG. 27 is a plan view of a tri-level ruled array of cube corners of theinvention, including FIG. 26B;

FIG. 28 is a plan view of a tri-level ruled array of cube corners of theinvention with a 50°-60°-70°triangle and its centroid superimposed;

FIG. 29 is a perspective view of a portion of the cube corner array ofFIG. 28;

FIG. 30A illustrates a quasi-triangular cube corner of the inventionwith displacement parameter ρ=0.75;

FIG. 30B illustrates the eight apex locations for a ruling in which thegrooves alternate between ρ=0.5 depths and ρ=0.75 depths;

FIGS. 31A-31E are polar graphs of geometric efficiency over all ω with βbetween 0° and 60°, for cube corner arrays of the invention,illustrating the improvement from ρ=0 through ρ=0.75;

FIG. 32A and 32B are graphs derived from FIGS. 31A-31E, FIG. 32Adetailing the geometric efficiency for the two ω's −90° and +90°, andFIG. 32B for the two ω's 0° and 180°;

FIG. 33 shows schematically a two pin assembly using 50°-60°-70° ρ=0.5cube corners of the invention;

FIG. 34 shows schematically a two pin assembly using mirror images ofthe cube corners shown in FIG. 33;

FIG. 35 graphs the geometric efficiency, as polycarbonate prisms, ofpinned cube corner designs of the invention, like those of FIGS. 33 and34, for the range of ρ's from 0 to 1;

FIG. 36A is a polar graph illustrating the geometric efficiency, aspolycarbonate prisms, of the two-pin structures of FIGS. 33 or 34;

FIG. 36B is a polar graph illustrating the geometric efficiency, aspolycarbonate prisms, of the four-pin structure that combines FIGS. 33and 34;

FIG. 37 is a plan view of a single quasi-triangular cube corner of thepreferred embodiment;

FIG. 38 is a plan view of a ruled array of the cube corners of FIG. 37;

FIG. 39 is a Chandler diagram for the two rotations of the cube cornerillustrated in FIG. 38;

FIG. 40 is a polar graph of geometric efficiency over all ω with βbetween 0° and 60° for the preferred embodiment;

FIGS. 41A and 41B are graphs illustrating the geometric efficiency ofthe preferred embodiment with respectively, groove g3 vertical in thearticle and groove g3 rotated 10°;

FIGS. 42A and 42B show schematically four pin assemblies of thepreferred embodiment, the first with groove g₃ 10° off the vertical orthe horizontal, the second with groove g₃ vertical or horizontal;

FIGS. 43A and 43B illustrate Chandler diagrams corresponding to the pinsof FIGS. 42A and 42B;

FIG. 44 is a graph illustrating the geometric efficiency of the pinassemblies schematized in FIGS. 42A and 42B;

FIGS. 45A and 45B are graphs illustrating calculated values of thecoefficient of retroreflection of the preferred embodiment.

DETAILED DESCRIPTION OF THE INVENTION

Two geometrical theorems are useful for explaining the invention.

Theorem 1 (FIG. 23). For any three points A, B, and C in a plane thereis one and only one point O above the plane such that AO, BO, and CO aremutually perpendicular. Furthermore, a perpendicular from O to the planefinds the orthocenter of triangle ABC, that is, point H where the threealtitudes of triangle ABC meet.

Theorem 2 (FIG. 24). In a plane, if ABC is any triangle and A′B′C′ isthe same triangle rotated 180° about some point in the plane, then thegreatest possible area of intersection of the two triangles is ⅔ thearea of one triangle, and this occurs only when the rotation is aboutthe centroid of triangle ABC, that is, about the point G where the threemedians of triangle ABC meet. The centroid of triangle A′B′C′ is at thesame point.

Theorem 1 implies that whenever a triangular cube corner is viewednormal to the plane of the triangle, the cube corner's three dihedraledges appear as the three altitudes to the triangle and the cubecorner's apex appears at the triangle's orthocenter. For ruledtriangular cube corners this view corresponds to β=0° illumination.

Theorem 2 implies that in order for ruled triangular cube corners tohave large effective aperture for β=0° illumination, its apex shouldappear at the triangle's centroid when viewing normal to the plane ofthe triangle.

For non-equilateral triangles, the orthocenter is distinct from thecentroid. FIG. 25 shows orthocenter H quite far from centroid G for atriangle corresponding to a compound canted triangular cube corner. ThusTheorem 2 says the cube corner apex should be located somewhere Theorem1 says it won't be.

The cube corner arrays of this invention are rulable, that is, they canbe generated by the repeated straight-line motion of shaped tools alongpaths parallel to a common plane, called the ruling plane. In particularthe tools cut symmetrical vee-grooves defining the cube corner arrays ofthis invention.

Points ABCH of FIG. 26A, corresponding to those of plane FIG. 25,illustrate a solid cube corner with apex H. Point G of FIG. 25,reappears in FIG. 26A as a possible new location for the apex. FIG. 26Aillustrates schematically how a cube corner apex might be displacedwhile the three cube corner faces remain parallel to their threeoriginal planes. The meeting of dihedral edges at new apex H looksidentical to the meeting of dihedral edges at original apex G. Thecompound cant has been preserved. The angles A, B, and C of triangle ABCare respectively 76.9°, 43.9°, and 59.2°. The same approximately 103.8°vee-groove cutter that formed face HBC can form a face like schematicGKL by cutting to a lesser depth. The same approximately 43.7°vee-groove cutter that formed face HCA can form a face like schematicGLCAJ by cutting to a greater depth. The same approximately 58.9° cutterthat formed face HAB can form a face like schematic GJBK by cutting toan intermediate depth.

FIG. 26B illustrates how nearly the schematic FIG. 26A is realized bysuch tri-level ruling. Regarding length CB as 1 and the apex G as atdepth 0 in FIG. 26B, the deepest, sharpest vee-groove, shown along CA,has depth 0.809, the shallowest, bluntest vee-groove, shown along OL,has depth 0.181, and the intermediate vee-groove, shown along JM, hasdepth 0.464. The deepest, sharpest groove forms apparently 5-sided faceGLCAJ as in FIG. 26A. The shallowest bluntest vee-groove does not formtrilateral face GKL as in FIG. 26A, but rather a quadrilateral faceGNOL. The intermediate depth, intermediate width vee-groove does notform quadrilateral face GJBK as in FIG. 26A, but rather quadrilateralface GJMN. In addition there is an apparently quadrilateral facet MNOB.This facet does not belong to the cube corner. It was formed by theopposite half of the sharpest, deepest vee-groove cutter that formedface GLCAJ when it made a different groove, namely that at point B.

For the cube corner with apex G in FIG. 26B, each of the cube faces hastwo dihedral edges and one or more non-dihedral edges. It is seen thatthe longest edge of each cube faces, respectively CA, OL, JM, is anon-dihedral edge along the root of the vee-groove defining that face.The expression “long face edges” of a cube corner refers to the longestedges of each of the faces.

FIG. 27 eliminates the artificiality of viewing the cube corner of FIG.26B as triangular by showing it within a ruled array. None of thetriangle's vertices A, B, C are distinguished points in the plan view ofthe array. Triangle ABC may be located in the array, and associated withthe cube corner having vertex G, but the face having angle LGJ at theapex extends beyond the triangle ABC, and an area MNOB of the trianglecannot participate in retroreflection by the G cube corner but doesparticipate, although weakly, in retroreflection of illumination withcertain β and ω, by the neighbor cube having vertex G′. The cube cornersof this invention are not strictly triangular, and are called“quasi-triangular”.

The inventive cube corners differ from those of Mimura et al. U.S. Pat.Nos. 6,083,607, 6,318,866 B1, and 6,390,629 B1 in having compound cant,rather than emp or fmp cant. In consequence no two of the inventive cubecorner's faces are congruent and no two of the defining groove depthsagree.

The cube corner design of FIG. 26B and FIG. 27 has a very large 17.6°compound cant and is given primarily to illustrate the inventive conceptof apex displacement by tri-level ruling. FIG. 28 illustrates a similarapplication of the method of apex displacement to the more practical9.74° canted cube corner illustrated in FIGS. 13A, 14, 18C, and 20B. Thegrooving directions were at 50°, 60°, and 70° to one another, and thegrooving depths were chosen to displace the cube apex. FIG. 28 shows thecube corner array and includes a superimposed 50°-60°-70° triangle withits medians showing that the apex was displaced to the centroid of thetriangle.

FIG. 28 also shows the directions of each of the cube axes. Opposingneighbors are adjacent and have faces that are opposite sides of onevee-groove. The axis of each cube corner is skew to the axes of itsthree opposing neighbors.

FIG. 29 is a perspective view of a portion of the same cube cornerarray. The deepest groove is seen on the left side, the intermediategroove on the right side, and the shallowest groove is seen up themiddle of FIG. 29. FIG. 28, like all the plan views, is a view into acube corner, so it is a view through the flat underside of the solidillustrated in FIG. 29.

By suitable tri-level ruling the apex can be displaced to anywhere inthe triangle. In the present invention, the displacement is toward thecentroid, but not necessarily to the centroid. In a plan view of atriangular cube corner of the desired compound cant, such as FIG. 25, astraight line may be drawn connecting the orthocenter H and centroid G.The family of designs in which the apex is displaced along the line fromH to G is parameterized by the fraction ρ. ρ=0 describes the prior artdesign with no apex displacement. ρ=1 describes the design with apexdisplaced to the triangle's centroid. It has been found thatdisplacements in the range ρ=0.25 to ρ=0.75 are preferred for thepreferred compound cants. Also it has been found that displacing theapex to a point Q off the line HG has no practical advantage ordisadvantage over displacing it to a point that is on the line HG andnear Q.

FIG. 30A shows the ρ=0.75 quasi-triangular cube corner based on the50°-60°-70° triangle. In space, groove roots g₁, g₂, and g₃, being atthree different levels, have no intersections, even when extended. Butin the plan view of FIG. 30A, the extended roots form the same50°-60°-70° triangle as they would have formed if they were of equaldepth. Point H is the orthocenter, and point G the centroid, of thistriangle. In the plan view, the cube corner apex X is seen to bedisplaced 75% of the way from H to G. ρ=HX/HG=0.75.

The Chandler diagram for the cube corner prism of FIG. 30A is given,with suitable rotation, in FIG. 13B if the prism is made of acrylic or,with suitable rotation, in FIG. 20B if the prism is made ofpolycarbonate. The rotations are necessary because in FIG. 30A, thelongest side of the triangle is 40° counterclockwise of vertical, whilein FIG. 13A it is vertical and in FIG. 20B it is 10° counterclockwise ofvertical. The Chandler diagram is independent of apex displacement.

Effective aperture depends on apex displacement. FIGS. 31A-E show howthe geometric efficiency of 50°-60°-70° triangular and quasi-triangularcube corner prisms made of polycarbonate depend on the apex displacementparameter ρ. The geometric efficiency is calculated for a ruled arrayincluding two prism orientations 180° apart. The triangles are orientedas shown in FIG. 20B, or rotated 180°. The triangles all have theirlongest side 10° counter-clockwise of vertical. In each of FIGS. 31A-Ethe radial direction represents entrance angle β from 0° to 60°, whilethe circumferential direction represents orientation angle ω. Thus FIGS.31A-E have the same format as Chandler diagrams except for thelimitation in entrance angle.

Geometric efficiency is the product of the effective aperture andcombined face reflectance. Combined face reflectance is slightly moreinformative than the Chandler diagram, because the latter only showswhen the combined face reflectance is 100%. Geometric efficiency mayconveniently be determined by ray tracing methods which incorporateFresnel reflection at surfaces. It may also be determined by separatelycalculating the effective aperture by the method of FIG. 2A and thecombined face reflectance by applying the Fresnel equations to all threeinternal reflections. The latter calculation is made simpler by therecognition that the incidence angle which an illumination ray makes ona any face in a cube corner, whether met first, second, or third, isequal to the incidence angle which a ray parallel to the incoming rayand meeting that face first, makes at that face. Polarization is ignoredin the determination of geometric efficiency.

The five FIGS. 31A, 31B, 31C, 31D, 31E are for ρ=0, 0.25, 0.5, 0.75, 1,respectively. The first three figures show the progressive improvementin geometric efficiency from ρ=0 of the prior art to ρ=0.25 and then toρ=0.5. The next figure suggests slight additional improvement withρ=0.75. The last figure shows decline with ρ=1. That is, ρ=1 isinsignificantly better than ρ=0.75 at small β, but significantly worseat large β.

FIGS. 31A-E illustrate the importance of the rotation of the prismaticelements for achieving good entrance angularity for all four omegas:−90°; 0°; 90°; 180°. The 10° counter-clockwise rotation used for theseexamples represents a compromise in which geometric efficiency at verylarge entrance angles for omega=±90° is obtained at the expense ofgeometric efficiency at somewhat smaller entrance angles at omega=0° and180°. Other rotations between about 20° counter-clockwise and about 20°clockwise could well be chosen to effect other compromises.

FIGS. 32A-B show some of the data of FIGS. 31A-E for quantitativecomparison. FIG. 32A shows the geometric efficiencies on horizontalslices of FIGS. 31A-E, corresponding to omega −90° or +90°. FIG. 32Bshows the geometric efficiency on vertical slices of FIGS. 31A-Ecorresponding to omega 0° or 180°. The great improvement over ρ=0 cubecorners of prior art is evident. The advantage of ρ=0.75 over ρ=1 isalso evident. ρ=0.5 is generally advantageous over ρ=0.25. ρ=0.75 andρ=0.5 are the most virtuous of the five apex displacements considered.The choice between ρ=0.75 and ρ=0.5 depends on the relative importanceof small and large entrance angles, respectively.

Since performance is virtuous over a range of apex displacements ρ, andsince small distance of the apex from the HG line on which ρ is definedis harmless, it is possible to apply the invention in such a way thatthe groove depths within each groove set are not held constant. Forexample, the groove depths in each set could alternate between the depththat would produce ρ=0.75 and the depth that would produce ρ=0.5. Inthis example, one-eighth of the cube corners are ρ=0.75 versions andone-eighth are ρ=0.5 versions. The remaining three-fourths consists ofsix hybrid types with their apices not on the HG line. FIG. 30B shows anenlargement of the HG line from FIG. 30A including the ρ=0.75 point X.Point Y in FIG. 30B is the ρ=0.5 point. The six smaller points show theother apex locations for the example ruling having alternating depths ineach groove set.

While the cube corner designs characterized in FIGS. 31B-D and 32A-Bhave good entrance angularity for all four omegas: −90°; 0°; 90°; 180°;the directions are not equivalent. No cube corner prism together withits 180° rotated mate can achieve that. For applications where equalperformance is required in all four directions the old trick of“pinning”, lately termed “tiling”, may be employed. The mechanics ofpinning was disclosed in Montalbano U.S. Pat. No. 4,460,449, and pinningwas used in the Stimsonite “High Performance Grade” prismatic sheetingfirst marketed in 1986. Large thick ruled masters bearing identical cubecorner arrays were diced into square pins, which were reassembled, withrotations of the cube corner array, into a new large thick master. Theresult was cube corner prisms having more than two rotations in thefinal array. FIG. 33 shows schematically a two-pin assembly utilizingthe 50°-60°-70° ρ=0.5 cube corner prisms of the present invention. Theleft pin in FIG. 33 has the direction of its bluntest groove 10°counter-clockwise of vertical, the same rotation used for FIGS. 31A-E.The right pin in FIG. 33 has the same cube corners rotated by 90°clockwise, or equivalently counter-clockwise. The geometric efficiencyof this 10° and 100° pinned assembly is an average of the geometricefficiencies of the separate pins. FIG. 35 shows the geometricefficiency in all four main omega directions for this pinned structurefor each of ρ=0 (prior art), 0.25, 0.5, 0.75, and 1. FIG. 35 is exactlythe averaging of FIGS. 32A and 32B. FIG. 35 shows again that the ρ=0(prior art) and ρ=1 designs are the least effective. The ρ=0.25 designis somewhat inferior to the ρ=0.5. The choice between ρ=0.5 and ρ=0.75depends on application. FIG. 36A shows the geometric efficiency of theρ=0.5 design in this pinned structure. FIG. 36A is to be compared withFIG. 31C which used the same prisms without pinning. FIG. 36A has 90°rotational symmetry, but it still lacks left-right symmetry. FIG. 34shows schematically two pins with 70°-60°-50°cube corners which aremirror images of the 50°-60°-70° cube corner pins shown schematically inFIG. 33. The geometric efficiency of the FIG. 34 structure will be themirror image of FIG. 36A. Assembling all four pins from FIGS. 33 and 34into a single sheeting structure results in the geometric efficiencyshown in FIG. 36B which has both 90° rotational symmetry and left-rightsymmetry. The graphs of FIG. 35 apply to the four pin structure.

While it achieves symmetry, a pinned assembly also has disadvantages.Accurate pins with ruled ends are difficult to make and to assemble.There is inevitable loss of retroreflectance at pin edges where cubesare truncated or distorted. It is realistic to derate the efficienciesin FIGS. 35, 36A-B by 0.95× for these reasons.

The table below gives ruling dimensions for 50°-60°-70° cube corners atρ=0.25, 0.5, 0.75, and 1, with ρ=0 of the prior art included forreference. Since the depth of the structure increases with increasingrho, there is manufacturing advantage to smaller rho.

TABLE 1 50°-60°-70° cube corners; triangle area = 1 groove groove groovedepth below apex vee spacing ρ = 0 ρ = 0.25 ρ = 0.5 ρ = 0.75 ρ = 1 g₁54.57° 1.458 0.594 0.681 0.768 0.855 0.942 g₂ 67.10° 1.289 0.594 0.6070.621 0.635 0.648 g₃ 88.22° 1.188 0.594 0.548 0.501 0.455 0.409 The twobluntest grooves, g₂ and g₃, meet at the smallest angle of the triangle,50°. The two sharpest grooves, g₁ and g₂, meet at the largest angle ofthe triangle, 70°.

A Preferred Embodiment

All the 50°-60°-70° cube corner prisms of polycarbonate have a suddenfall in geometric efficiency at about β=11° for ω=±90°. This is causedby the loss in TIR in half the cube corners as shown in the Chandlerdiagram in FIG. 20B. The rather large 9.74° compound cant isresponsible. Designs that have all cubes functioning to at least β=15°for all ω are preferable for road sign applications. A 53°-60°-67° cubecorner design having 6.70° compound cant is used in this preferredembodiment of the present invention. Table 2 gives ruling dimensions for53°-60°-67° cube corners at ρ=0.25, 0.5, 0.75, and 1, with ρ=0 of theprior art included for reference.

TABLE 2 53°-60°-67° cube corners; triangle area = 1 groove groove groovedepth below apex vee spacing ρ = 0 ρ = 0.25 ρ = 0.5 ρ = 0.75 ρ = 1 g₁59.35° 1.413 0.608 0.662 0.717 0.772 0.827 g₂ 68.88° 1.303 0.608 0.6140.621 0.627 0.633 g₃ 82.54° 1.226 0.608 0.572 0.537 0.501 0.466

For this embodiment, the cube apex is displaced by ρ=0.5. The prismmaterial is polycarbonate with n=1.586. The cube corner array is rotatedso the triangles' longest sides, corresponding to the cube cornersbluntest, shallowest grooves, denoted g₃ in the TABLE 2, are 10°counter-clockwise from vertical.

FIG. 37 shows a single quasi-triangular cube corner of this preferredembodiment. FIG. 38 shows a portion of a ruled array of such cubecorners. FIG. 39 shows the Chandler diagram for a pair of such cubecorners.

The arms in the Chandler diagram for either cube corner in FIG. 39 makemutual angles of approximately 140.0°, 119.1°, 100.9°. For determiningthese angles we take the center point of the arm at β=50°. In themanufactured article the cube corners are rotated so that the arm whichmakes the two smaller angles is nearly aligned with the long sheet. Thiscauses one of the other arms to arms to be approximately 10° away fromthe perpendicular direction, while the third arm is approximately 30°away from the perpendicular direction.

The Chandler diagram in FIG. 39 shows unlimited entrance angle in the±90° directions and entrance angle limited to about 57° in the 0° and180° direction. Conservatively, useful entrance angle is limited to 45°to 50° degrees in the latter direction.

Most road sign applications involve angle β in the range 0° to 15°, withangle ω unrestricted. Successful road sign sheeting needs to have allcubes functioning within this range. The Chandler diagram FIG. 39 showsTIR over the full 15° circle. This TIR is sustained by 53°-60°-67° cubecorners provided the refractive index is at least 1.580.

The ρ=0 prior art array ruled with the angles given above, and withgrooves of equal depth, is comprised of triangular cube corners and hasgeometric efficiency 59.3% for normal incidence. The 59.3% value shouldbe compared to the 66.7% geometric efficiency of equilateral, uncantedcube corners. In this preferred ρ=0.5 embodiment, the apex is displacedhalf way to the centroid. The displacement is accomplished by makingquite unequal groove depths while leaving all groove angles, directionsand spacings unchanged. From TABLE 2, the shallowest groove has 88% ofthe depth of the grooves of the corresponding prior art array, theintermediate groove has 102% of the prior art depth, and the deepestgroove has 118% of the prior art depth.

The resulting array of trilevel quasi-triangular cube corners hasgeometric efficiency 64.8% at normal incidence, 1.09× as great as forthe single depth triangle cube corners. The gain diminishes somewhatwith increasing entrance angle, but parity isn't reached until between50 and 55 degrees.

FIG. 40 shows the geometric efficiency of this preferred embodiment overthe full range of orientation angle ω, for entrance angle β between 0°and 60°. FIG. 41A graphs data from horizontal and vertical slices ofFIG. 40. From FIG. 41A, at β=50°, the embodiment has geometricefficiency 24.8% in the ±90° omega directions and 16.5% in the 0° and180° omega directions.

Applicant knows of no other rulable cube corner array consisting oftriangles or quasi-triangles in n=1.586 material that combines the threedescribed features of this preferred embodiment:

-   -   1. All cubes having TIR at β=15° for all ω.    -   2. Over 64% geometric efficiency at β=0°.    -   3. Over 16% geometric efficiency at β=50° for ω=−90°, 0°, 90°,        180°.

It will be understood that this preferred embodiment includes designcompromises. For example, without the 10° rotation, FIG. 41B would takethe place of FIG. 41A. Comparison of FIGS. 41A and 41B shows that the10° rotation substantially improved the geometric efficiency for β=50°in the 0° and 180° omega directions, but it substantially reduced thegeometric efficiency for β=35° in the 0° and 180° omega directions.Users of this invention will appreciate that retroreflector design ofteninvolves difficult compromises, often involving β and ω tradeoffs. Asdescribed above, use of four pins can symmetrize the performance. FIG.42A shows the representative cube corners, with 10° rotation, in each ofthe four pins. FIG. 42B shows the representative cube corners, with norotation, in each of the four pins. For four-pinned assemblies, the twocurves of FIG. 41A become a single averaged curve, and also the twocurves of FIG. 41B become a single averaged curve. FIG. 44 shows the tworesulting curves of geometric efficiency. After the pinning, the twodesigns are more nearly equivalent, but some β=35° versus β=50°tradeoffs remain. Note that the geometric efficiency curves of FIG. 44do not take into account the estimated 0.95× derating for pinninglosses. Note also that the data for these comparisons was collected atintervals of 5° in β and of 10° in ω. FIGS. 43A and 43B show theChandler diagrams corresponding to each of the pins shown in FIGS. 42Aand 42B.

The cube corners of this invention are expected to be micro-cube cornerswhich in the present state of the art are best formed by ruling. Theroots of the three grooves defining each quasi-triangular cube corner ofthis invention generally have no intersections, but lines along thegroove roots determine a triangle when viewed in plan in a directionperpendicular to the ruling plane. The area of this triangle is lessthan about 0.3 mm² for micro-cube corners. Typically, the area of thetriangle is in the range from about 0.007 mm² to about 0.07 mm².

Two optical factors remain for describing an embodiment: the size of theoptical elements and the elements' aberrations. Size must be specifiedbecause cube corner prisms in sheetings are small enough that theirdiffraction characteristics significantly affect retroreflectance.Geometric efficiency depends the effective aperture area relative to thewhole structure area. Diffraction efficiency depends on the absolutearea and the shape of the effective aperture. Element size andaberration should be chosen according to the intended application of thesheeting. The cube corners of the present invention are intended forroad sign and similar sheeting applications. Road sign sheetingprimarily for long distance retroreflection will have relatively largeprisms, with triangle areas greater than 0.03 mm², and small aberrationsin order to produce a tight retroreflected light beam, perhaps specifiedto just 0.5° of divergence. Road sign sheeting intended for shortdistance retroreflection should have intentional aberrations forspreading the retroreflected light into a wider beam, perhaps specifiedto 2.0° divergence. It is then convenient and economical to make theprisms rather small, with triangle areas less than 0.015 mm², and letdiffraction do some of the spreading. This precludes an intentionalreduction at the center of the beam, but there is little to be gainedfrom that strategy. The expected levels of retroreflected beam intensityacross a 2.0° divergent beam are ¼ the levels across a 0.5° divergentbeam of similar form. Thus designing a short distance road sign sheetingis much more challenging than designing a long distance road signsheeting. While the high geometric efficiency of the cube corners of thepresent invention make them suitable for short distance road signsheeting, this preferred embodiment is a medium distance road signsheeting, intended for the majority of road signs

This preferred embodiment is a ruling of quasi-triangular 53°-60°-67°cube corners, with axis displacement ρ=0.5, and triangle area 0.015 mm².The triangle area being 0.015 mm² implies that all the unitless lineardimensions in TABLE 2 must be multiplied by √{square root over (0.015mm²)}=0.1225 mm to give the ruling dimensions in TABLE 3.

TABLE 3 groove groove groove vee spacing depth g₁ 59.35° 0.173 mm 0.088mm g₂ 68.88° 0.160 mm 0.076 mm g₃ 82.54° 0.150 mm 0.066 mm

Users of the invention will do more accurate calculation for groove veeand groove spacing than given in TABLE 3. However, groove depth is notcritical. Error of 0.001 mm in groove depth leads to small error in ρ orsmall deviation of the apex from the HG line, which can be tolerated.

Methods for introducing cube corner aberrations during ruling arewell-known in the art from Appeldorn U.S. Pat. No. 4,775,219. PendingU.S. patent application Ser. No. 10/167,135, filed Jun. 11, 2002,claiming the benefit of Ser. No. 60/297,394, filed Jun. 11, 2001,discloses other methods. Yet another method for introducing cube corneraberrations during the tool making process, disclosed in a pending U.S.patent application filed Dec. 12, 2002 entitled “Retroreflector withControlled Divergence Made by the Method of Localized Substrate Stress”,is employed in the present preferred embodiment. The sheeting is assumedto be embossed in clear colorless polycarbonate by the methods disclosedin Pricone et al. U.S. Pat. No. 4,486,363, using tools made by methodsdisclosed in Pricone U.S. Pat. No. 4,478,769 and containing the cubecorners described by TABLE 3 as modified with aberrations similar inscale and distribution to those disclosed in the Dec. 12, 2002application. Specifically, the average dihedral angle is 1.0 arc minutesand the standard deviation is 7.3 arc minutes.

At the time of this application there are only calculated coefficientsof retroreflection, R_(A), for a sheeting. The calculations include theeffects of polarization and diffraction in accordance with the methodsof Edson R. Peck (“Polarization Properties of Corner Reflectors andCavities”, Journal of the Optical Society of America, Volume 52, Number3, March, 1962). Retroreflection involving multiple cubes is ignored. AnR_(A) reduction of 30% is allowed for the non-retroreflecting area ofwelds in non-metallized sheeting and an additional 10% is allowed fordefects in tooling, materials and manufacturing.

FIGS. 45A and 45B show the calculated coefficient of retroreflection ofthis preferred embodiment as it would be measured according to ASTM E810-02 Standard Test Method for Coefficient of Retroreflection ofRetroreflective Sheeting Utilizing the Coplanar Geometry. FIG. 45A isfor entrance angle β=4° and FIG. 45B is for entrance angle β=30°.

R_(A) measures the luminous intensity of the retroreflected-lightrelative to the perpendicular illuminance provided by the illuminationbeam, for one square meter of sheeting. The ASTM test method measuresthis intensity at various divergence angles α in the plane that containsthe illumination direction and the sheeting surface normal. Angle ε inthe ASTM E 810-02 test method is equal to orientation angle ω. Allretroreflection angles are explained in ASTM E 808-01 Standard Practicefor Describing Retroreflection. Angle ω of this application is denoted“ω_(s)” in the ASTM document.

Table 4, derived from FIGS. 45A and 45B, gives calculated R_(A) valuesof this preferred embodiment sheeting at the 16 angular points requiredin ASTM D 4956-02 Standard Specification Retroreflective Sheeting forTraffic Control.

TABLE 4 α β₁ β₂ ε R_(A) 1 0.1° −4° 0° 0° 917 2 0.1° −4° 0° 90°  859 30.1°   30°  0° 0° 594 4 0.1°   30°  0° 90°  437 5 0.2° −4° 0° 0° 561 60.2° −4° 0° 90°  493 7 0.2°   30°  0° 0° 376 8 0.2°   30°  0° 90°  329 90.5° −4° 0° 0° 377 10 0.5° −4° 0° 90°  422 11 0.5°   30°  0° 0° 276 120.5°   30°  0° 90°  139 13 1.0° −4° 0° 0° 103 14 1.0° −4° 0° 90°  95 151.0°   30°  0° 0° 82 16 1.0°   30°  0° 90°  45

Those skilled in the art will appreciate that the above preferredembodiment is only illustrative of the invention, and that manymodifications and variations are possible without departing from thescope and spirit of the invention. It is expected that road signs madefrom sheetings made according to the invention will efficientlyretroreflect vehicle head lights at night and be bright from sun andskylight by day.

1. A rulable array of cube corners each of which has its long face edgesmutually skew.
 2. The array defined in claim 1, said cube corners havingcompound cant.
 3. The array defined in claim 2, said cube corners beingmale and being formed in a transparent material having a secondsubstantially flat surface, said cube corners being retroreflective oflight entering the second surface.
 4. The array defined in claim 1, theaxis of each said cube corner being skew to the axes of its threeopposing neighbors.
 5. The array of micro-cube corner elements definedin claim 4, wherein each two micro-cube corners are geometricallycongruent.
 6. An array of micro-cube corner elements defined by threesets, S1, S2, S3, of symmetrical vee-grooves such that within each setthe grooves are parallel and equidistant, and such that there is areference plane such that the roots of the grooves in S1 are generallyfarther from the reference plane than the roots of the grooves in S2which are generally farther from the reference plane than the roots ofthe grooves of S3.
 7. The micro-cube corner elements defined in claim 6,wherein no two pairs of lines aligned with the roots of said three setsof vee-grooves makes the same angle when said array of micro-cube cornerelements are viewed in plan.
 8. An array of micro-cube corners for whichthe apices lie in one plane and the long face edges lie in three planes.9. The array defined in claim 8, wherein the four mentioned planes areparallel, and the plane of one set of long face edges is no more than90% as far from the plane of apices than the plane of a second set oflong face edges, which in turn is no more than 90% of the distance fromthe plane of apices than the third set of long face edges.
 10. An arrayof cube corners defined by three sets of parallel vee-grooves, thedirections of said three groove sets making three angles no two of whichare equal, wherein, when viewed in plan in a direction normal to thearray, lines along the roots of said grooves determine a pattern oftriangles in which the apices of said cube corners lie at distances fromtheir respective triangle's centroid that is substantially less than thedistance between said triangle's orthocenter and its centroid.
 11. Thearray defined in claim 10, said cube corners being micro-cube corners.12. A retroreflective sheeting product comprising the cube corner arraydefined in claim 11, wherein the triangle areas are 0.1 mm² or smaller.13. The array defined in claim 11, said cube corners being male andbeing formed in a transparent material having a second substantiallyfiat surface, said cube corners being retroreflective of light enteringthe second surface.
 14. The array of retroreflective cube cornersdefined in claim 13 wherein the distance from apex to centroid is nogreater than 75% of said distance from orthocenter to centroid.
 15. Aretroreflective cube corner array defined in claim 13, wherein each ofsaid corner cube elements is canted between 4° and 16°.
 16. The array ofretroreflective cube corners defined in claim 13 wherein the directionsof groove sets make angles different from one another by at least 5°.17. The array of retroreflective cube corners defined in claim 16wherein the distance from apex to centroid is no greater than 75% ofsaid distance from orthocenter to centroid.
 18. The retroreflective cubecorner array defined in claim 17, wherein said cube corner elementsgenerally have effective aperture at least 60% for normally incidentillumination.
 19. The retroreflective cube corner array defined in claim17, wherein an index of refraction of said transparent material isbetween 1.40 and 1.80.
 20. An array of retroreflective cube cornerscomprised of a plurality of distinguishable sub-arrays, at least two ofwhich are as defined in claim
 17. 21. The retroreflective cube cornerarray defined in claim 20 in which the Chandler diagrams for some of thesub-arrays are identical to the Chandler diagrams for other of thesub-arrays except for a 90° rotation.
 22. The retroreflective cubecorner array defined in claim 20 in which the Chandler diagrams for someof the sub-arrays are identical to the Chandler diagrams for other ofthe sub-arrays except for being mirror images.
 23. The retroreflectivecube corner array defined in claim 17, wherein the index of refractionof said transparent material is at least 1.58 and wherein all cubecorner elements retroreflect with three total internal reflections forall entrance angles β up to 15° for all orientation angles ω.
 24. Theretroreflective cube corner array defined in claim 23, wherein said cubecorner elements have effective aperture at least 64% for normallyincident illumination.
 25. The retroreflective cube corner array definedin claim 24 wherein geometric efficiency is at least 16% at β=50° forfour values of orientation angle ω at separations of 90°.
 26. Aretroreflective sheeting product comprising the cube corner arraydefined in claim 25 and having the two of the four said orientationangles ω aligned with the web direction of the sheeting.